| Title:
|
On universality of semigroup varieties (English) |
| Author:
|
Demlová, Marie |
| Author:
|
Koubek, Václav |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
42 |
| Issue:
|
4 |
| Year:
|
2006 |
| Pages:
|
357-386 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A category $K$ is called $\alpha $-determined if every set of non-isomorphic $K$-objects such that their endomorphism monoids are isomorphic has a cardinality less than $\alpha $. A quasivariety $Q$ is called $Q$-universal if the lattice of all subquasivarieties of any quasivariety of finite type is a homomorphic image of a sublattice of the lattice of all subquasivarieties of $Q$. We say that a variety $V$ is var-relatively alg-universal if there exists a proper subvariety $W$ of $V$ such that homomorphisms of $V$ whose image does not belong to $W$ contains a full subcategory isomorphic to the category of all graphs. A semigroup variety $V$ is nearly $J$-trivial if for every semigroup $S\in V$ any $ J$-class containing a group is a singleton. We prove that for a nearly $J$-trivial variety $V$ the following are equivalent: $V$ is $Q$-universal; $ V$ is var-relatively alg-universal; $V$ is $\alpha $-determined for no cardinal $\alpha $; $V$ contains at least one of the three specific semigroups. Dually, for a nearly $J$-trivial variety $V$ the following are equivalent: $V$ is $3$-determined; $V$ is not var-relatively alg-universal; the lattice of all subquasivarieties of $V$ is finite; $V$ is a subvariety of one of two special finitely generated varieties. (English) |
| Keyword:
|
semigroup variety |
| Keyword:
|
band variety |
| Keyword:
|
full embedding |
| Keyword:
|
$f\!f$-alg-universality |
| Keyword:
|
determinacy |
| Keyword:
|
$Q$-universality |
| MSC:
|
08B15 |
| MSC:
|
08C15 |
| MSC:
|
18B15 |
| MSC:
|
20M07 |
| MSC:
|
20M99 |
| idZBL:
|
Zbl 1152.20046 |
| idMR:
|
MR2283018 |
| . |
| Date available:
|
2008-06-06T22:48:48Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/108013 |
| . |
| Reference:
|
[1] Adams, M. E., Adaricheva, K. V., Dziobiak, W. and A. V. Kravchenko, A. V.: Some open question related to the problem of Birkhoff and Maltsev.Studia Logica 78 (2004), 357–378. MR 2108035 |
| Reference:
|
[2] Adams, M. E and Dziobiak, W.: $Q$-universal quasivarieties of algebras.Proc. Amer. Math. Soc. 120 (1994), 1053–1059. MR 1172942 |
| Reference:
|
[3] Adams, M. E and Dziobiak, W.: Lattices of quasivarieties of $3$-element algebras.J. Algebra 166 (1994), 181–210. MR 1276823 |
| Reference:
|
[4] Adams, M. E and Dziobiak, W.: Finite-to-finite universal quasivarieties are $Q$-universal.Algebra Universalis 46 (2001), 253–283. MR 1835799 |
| Reference:
|
[5] Adams, M. E and Dziobiak, W.: Quasivarieties of idempotent semigroups.Internat. J. Algebra Comput. 13 (2003), 733–752. MR 2028101 |
| Reference:
|
[6] Birjukov, A. P.: Varieties of idempotent semigroups.Algebra i Logika 9 (1970), 255–273. (in Russian) MR 0297897 |
| Reference:
|
[7] Clifford, A. H. and Preston, G.B.: The Algebraic Theory of Semigroups.AMS, Providence, (vol. 1 1961, vol. 2 1967). |
| Reference:
|
[8] Demlová, M. and Koubek, V.: Endomorphism monoids of bands.Semigroup Forum 38 (1989), 305–329. MR 0982011 |
| Reference:
|
[9] Demlová, M. and Koubek, V.: Endomorphism monoids in varieties of bands.Acta Sci. Math. (Szeged) 66 (2000), 477–516. MR 1804205 |
| Reference:
|
[10] Demlová, M. and Koubek, V.: Weaker universalities in semigroup varieties.Novi Sad J. Math. 34 (2004), 37–86. MR 2136462 |
| Reference:
|
[11] Demlová, M. and Koubek, V.: Weak alg-universality and $Q$-universality of semigroup quasivarieties.Comment. Math. Univ. Carolin. 46 (2005), 257–279. MR 2176891 |
| Reference:
|
[12] Dziobiak, W.: On subquasivariety lattices of some varieties related with distributive $p$-algebras.Algebra Universalis 21 (1985), 205–214. Zbl 0589.08007, MR 0835971 |
| Reference:
|
[13] Dziobiak, W.: The subvariety lattice of the variety of distributive double $ p$-algebras.Bull. Austral. Math. Soc. 31 (1985), 377–387. Zbl 0579.06012, MR 0801597 |
| Reference:
|
[14] Fennemore, Ch.: All varieties of bands.Semigroup Forum 1 (1970), 172–179. Zbl 0206.30401, MR 0271257 |
| Reference:
|
[15] Gerhard, J. A.: The lattice of equational classes of idempotent semigroups.J. Algebra 15 (1970), 195–224. Zbl 0194.02701, MR 0263953 |
| Reference:
|
[16] Gerhard, J. A. and Shafaat, A.: Semivarieties of idempotent semigroups.Proc. London Math. Soc. 22 (1971), 667–680. MR 0292967 |
| Reference:
|
[17] Goralčík, P. and Koubek, V.: Minimal group–universal varieties of semigroups.Algebra Universalis 21 (1985), 111-122. MR 0835975 |
| Reference:
|
[18] Hedrlín, Z. and Lambek, J.: How comprehensive is the category of semigroups?.J. Algebra 11 (1969), 195–212. MR 0237611 |
| Reference:
|
[19] Hedrlín, Z. and Pultr, A.: Relations (graphs) with finitely generated semigroups.Monatsh. Math. 68 (1964), 213–217. MR 0168684 |
| Reference:
|
[20] Hedrlín, Z. and Pultr, A.: Symmetric relations (undirected graphs) with given semigroups.Monatsh. Math. 69 (1965), 318–322. MR 0188082 |
| Reference:
|
[21] Hedrlín, Z. and Sichler, J.: Any boundable binding category contains a proper class of mutually disjoint copies of itself.Algebra Universalis 1 (1971), 97–103. MR 0285580 |
| Reference:
|
[22] Koubek, V.: Graphs with given subgraphs represent all categories.Comment. Math. Univ. Carolin. 18 (1977), 115–127. Zbl 0355.18006, MR 0457276 |
| Reference:
|
[23] Koubek, V.: Graphs with given subgraphs represent all categories II.Comment. Math. Univ. Carolin. 19 (1978), 249–264. Zbl 0375.18004, MR 0498229 |
| Reference:
|
[24] Koubek, V. and Radovanská, H.: Algebras determined by their endomorphism monoids.Cahiers Topologie Géom. Différentielle Catég. 35 (1994), 187–225. MR 1295117 |
| Reference:
|
[25] Koubek, V. and Sichler, J.: Universal varieties of semigroups.J. Austral. Math. Soc. Ser. A 36 (1984), 143–152. MR 0725742 |
| Reference:
|
[26] Koubek, V. and Sichler, J.: Equimorphy in varieties of distributive double $p$-algebras.Czechoslovak Math. J. 48 (1998), 473–544. MR 1637938 |
| Reference:
|
[27] Koubek, V. and Sichler, J.: On relatively universality and $Q$-universality.Studia Logica 78 (2004), 279–291. MR 2108030 |
| Reference:
|
[28] Pultr, A. and Trnková, V.: Combinatorial, Algebraic and Topological Representations of Groups, Semigroups and Categories.North Holland, Amsterdam, 1980. MR 0563525 |
| Reference:
|
[29] Rosický, J.: On example concerning testing categories.Comment. Math. Univ. Carolin. 18 (1977), 71–75. MR 0432730 |
| Reference:
|
[30] Sapir, M. V.: Varieties with a finite number of subquasivarieties.Sib. Math. J. 22 (1981), 168–187. Zbl 0491.08011, MR 0638015 |
| Reference:
|
[31] Sapir, M. V.: Varieties with countable number of subquasivarieties.Sib. Math. J. 25 (1984), 148–163. MR 0746951 |
| Reference:
|
[32] Sapir, M. V.: The lattice of quasivarieties of semigroups.Algebra Universalis 21 (1985), 172–180. Zbl 0599.08014, MR 0855737 |
| Reference:
|
[33] Schein, B.M.: Ordered sets, semilattices, distributive lattices and Boolean algebras with homomorphic endomorphism semigroups.Fund. Math. 68 (1970), 31–50. Zbl 0197.28902, MR 0272686 |
| Reference:
|
[34] Schein, B.M.: Bands with isomorphic endomorphism semigroups.J. Algebra 96 (1985), 548–565. Zbl 0579.20064, MR 0810545 |
| . |
